Editing Cartesian coordinate system (section)

{{Short description|Most common coordinate system (geometry)}}
{{Use dmy dates|date=December 2022}}
[[File:Cartesian-coordinate-system.svg|thumb|Illustration of a Cartesian coordinate plane. Four points are marked and labeled with their coordinates: {{nowrap|(2, 3)}} in green, {{nowrap|(−3, 1)}} in red, {{nowrap|(−1.5, −2.5)}} in blue, and the origin {{nowrap|(0, 0)}} in purple.]]

In [[geometry]], a '''Cartesian coordinate system''' ({{IPAc-en|UK|k|ɑːr|ˈ|t|iː|zj|ə|n}}, {{IPAc-en|US|k|a:r|'|t|i:|ʒ|ə|n}}) in a [[plane (geometry)|plane]] is a [[coordinate system]] that specifies each [[point (geometry)|point]] uniquely by a pair of [[real number]]s called ''coordinates'', which are the [[positive and negative numbers|signed]] distances to the point from two fixed [[perpendicular]] [[oriented line]]s, called ''[[coordinate line]]s'', ''coordinate axes'' or just ''axes'' (plural of ''axis'') of the system. The point where the axes meet is called the ''[[Origin (mathematics)|origin]]'' and has {{math|(0, 0)}} as coordinates. The axes [[direction (geometry)|directions]] represent an [[orthogonal basis]]. The combination of origin and basis forms a [[coordinate frame]] called the '''Cartesian frame'''.

Similarly, the position of any point in [[three-dimensional space]] can be specified by three ''Cartesian coordinates'', which are the signed distances from the point to three mutually perpendicular planes. More generally, {{math|''n''}} Cartesian coordinates specify the point in an {{math|''n''}}-dimensional [[Euclidean space]] for any [[dimension]] {{math|''n''}}. These coordinates are the signed distances from the point to {{math|''n''}} mutually perpendicular fixed [[hyperplane]]s.

[[File:Cartesian-coordinate-system-with-circle.svg|thumb|Cartesian coordinate system with a circle of radius 2 centered at the origin marked in red. The equation of a circle is {{nowrap|1=(''x'' − ''a'')<sup>2</sup> + (''y'' − ''b'')<sup>2</sup> = ''r''<sup>2</sup>}} where ''a'' and ''b'' are the coordinates of the center {{nowrap|(''a'', ''b'')}} and ''r'' is the radius.]]

Cartesian coordinates are named for [[René Descartes]], whose invention of them in the 17th century revolutionized mathematics by allowing the expression of problems of geometry in terms of [[algebra]] and [[calculus]]. Using the Cartesian coordinate system, geometric shapes (such as [[curve]]s) can be described by [[equation]]s involving the coordinates of points of the shape. For example, a [[circle]] of radius 2, centered at the origin of the plane, may be described as the [[set (mathematics)|set]] of all points whose coordinates {{math|''x''}} and {{math|''y''}} satisfy the equation {{math|1=''x''<sup>2</sup> + ''y''<sup>2</sup> = 4}}; the [[area]], the [[perimeter]] and the [[tangent line]] at any point can be computed from this equation by using [[integral]]s and [[derivative]]s, in a way that can be applied to any curve.

Cartesian coordinates are the foundation of [[analytic geometry]], and provide enlightening geometric interpretations for many other branches of mathematics, such as [[linear algebra]], [[complex analysis]], [[differential geometry]], multivariate [[calculus]], [[group theory]] and more. A familiar example is the concept of the [[graph of a function]]. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including [[astronomy]], [[physics]], [[engineering]] and many more. They are the most common coordinate system used in [[computer graphics]], [[computer-aided geometric design]] and other [[computational geometry|geometry-related data processing]].